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May  2016, 36(5): 2757-2779. doi: 10.3934/dcds.2016.36.2757

## Random attractor of stochastic Brusselator system with multiplicative noise

 1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, United States

Received  December 2014 Revised  August 2015 Published  October 2015

Asymptotic dynamics of stochastic Brusselator system with multiplicative noise is investigated in this work. The existence of random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimations than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the attracting regularity to be an $(L^2\times L^2,H^1\times H^1)$ random attractor.
Citation: Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [2] A. V. Babin and M. I. Vishik, Regular attractors of semigroups of evolutionary equations,, J. Math. Pures Appl., 62 (1983), 441. [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845. doi: 10.1016/j.jde.2008.05.017. [4] T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation,, Proc. R. Soc. Lond. A, 457 (2001), 2041. doi: 10.1098/rspa.2001.0819. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002). [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307. doi: 10.1007/BF02219225. [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. [8] F. Flandoli and B. Schmalfu$\beta$, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21. doi: 10.1080/17442509608834083. [9] M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Modles in Biology, Chemistry and Population Genetics,, Springer, (2012). doi: 10.1007/978-3-642-22664-9. [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London, 463 (2007), 163. doi: 10.1098/rspa.2006.1753. [11] L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems,, J. Chem. Physics, 48 (1968), 1695. [12] J. C. Robinson, Stability of random attractors under perturbation and approximation,, Journal of Differential Equations, 186 (2002), 652. doi: 10.1016/S0022-0396(02)00038-4. [13] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001). doi: 10.1007/978-94-010-0732-0. [14] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-5037-9. [15] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269. doi: 10.3934/dcds.2014.34.269. [16] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonl. Anal. A, 75 (2012), 3049. doi: 10.1016/j.na.2011.12.002. [17] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301. doi: 10.3934/dcds.2014.34.301. [18] W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonl. Anal. A, 84 (2013), 61. doi: 10.1016/j.na.2013.01.014.

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##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. [2] A. V. Babin and M. I. Vishik, Regular attractors of semigroups of evolutionary equations,, J. Math. Pures Appl., 62 (1983), 441. [3] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, J. Differential Equations, 246 (2009), 845. doi: 10.1016/j.jde.2008.05.017. [4] T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation,, Proc. R. Soc. Lond. A, 457 (2001), 2041. doi: 10.1098/rspa.2001.0819. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS Colloquium Publications, (2002). [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307. doi: 10.1007/BF02219225. [7] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365. doi: 10.1007/BF01193705. [8] F. Flandoli and B. Schmalfu$\beta$, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21. doi: 10.1080/17442509608834083. [9] M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Modles in Biology, Chemistry and Population Genetics,, Springer, (2012). doi: 10.1007/978-3-642-22664-9. [10] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London, 463 (2007), 163. doi: 10.1098/rspa.2006.1753. [11] L. Prigogine and R. Lefever, Symmetry-breaking instabilities in dissipative systems,, J. Chem. Physics, 48 (1968), 1695. [12] J. C. Robinson, Stability of random attractors under perturbation and approximation,, Journal of Differential Equations, 186 (2002), 652. doi: 10.1016/S0022-0396(02)00038-4. [13] J. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge Univ. Press, (2001). doi: 10.1007/978-94-010-0732-0. [14] R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002). doi: 10.1007/978-1-4757-5037-9. [15] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 269. doi: 10.3934/dcds.2014.34.269. [16] Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems,, Nonl. Anal. A, 75 (2012), 3049. doi: 10.1016/j.na.2011.12.002. [17] Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 301. doi: 10.3934/dcds.2014.34.301. [18] W. Zhao, $H^1$-random attractors for stochastic reaction-diffusion equations with additive noise,, Nonl. Anal. A, 84 (2013), 61. doi: 10.1016/j.na.2013.01.014.
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